Mishnayos maseches Kinim can be reduced to a few rules. I think they're as follows (though it's been a while, so I may be messing some of these up).

A bird ola is brought on the top of the mizbeach, and chatas on the bottom. Do either one wrong, and it's invalid.

A chova is one chatas and one ola; a neder or n'dava is two olos. If you bring more olos or chataos than that from a ken, the excess are invalid.

Both birds of a ken are b'ne yona, or both torim. If a mixed ken was brought, only the chatas is valid; ben Azay says only the first brought is.

If a bird may have been designated as a chatas, it cannot be brought as an ola; and vice versa.

If a bird can't be brought or was brought but is invalid, bring a new one in its stead, and its former mate is fine.

Why, then, does the maseches go on and on with examples instead of simply stating the rules? Mishnayos, from what I understand, are usually written to be easy to memorize. Surely Kinim as we have it is harder to memorize than the above.

Sure it does. In 3:6 it says (in a certain case) that she has to bring Od Preidah Achat which means one bird. You're thinking of if she was noder a kan which is two birds and is the first case in that mishna.
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Double AA♦Jan 24 '12 at 5:41

Also, I think you mixed up Tanna Kama and Ben Azai. See 2:5
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Double AA♦Jan 24 '12 at 5:41

@DoubleAA "Od p'reda achas" is not talking about someone who made a n'dava of a single bird. And yes, I confused ben Azay and the t"k: thanks! I've corrected it. (As I said, it's been a while....)
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msh210♦Jan 24 '12 at 5:45

2 Answers
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While your summary might work for the general behavior of bringing sacrifices of birds in ideal situation, your summary does not cover, or give reference to the border cases, and what exactly is to be done with various birds or situations that can not be offered.

Also, at the point in time in which the mishna was written, these cases were not done in practice, so it seems that in these situations the Mishna is acting as it's own gemorah explaining as much as it can in as short a space as possible. The rules as you have written them , might have allowed for expansion, but that expansion would have been lost due to a lack of familiarity and practice. Just skimming through the mishnayot I see lots of information that is not covered in your short summary.

In addition, many people are able to learn extra idea and lessons from the Mishna which do not directly relate to the halachot as summerized. For example, you can learn some insights into childbirth and oaths from the mishnayot which one would not be able to learn from a summary of rules.

One significant rule you didn't mention is that if a bird from a group of "undesignated pairs" flies into another such group, it invalidates one bird in each group. Another important topic is the discussion at the end of the masechta regarding a kohen who didn't do what he was supposed to.

In any event, aside from the fact that there are more rules than the ones you mentioned, it should be noted that Mishnayos in general deal with practical, real-life cases rather than dry legal principles. For example, you will learn about the law when an ox gores a cow, or when two people fight over a tallis, rather than learning the bare rules that underlie those laws.

The two rules you mention in your first paragraph are just examples of the rules I give.
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msh210♦Jan 24 '12 at 5:34

How so? Perhaps an offending bird can be botul b'rov. At the very least you would need to add another rule, that birds are not botul. This is actually one of the major "themes" that run through the masechta.
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DaveJan 24 '12 at 13:49

Fair enough. (But my point wasn't which rules there are, only that there are few.)
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msh210♦Jan 24 '12 at 15:19

That's already 6 major rules, and there are many other details that haven't been mentioned. Remember that the entire masechta is only 3 chapters long!
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DaveJan 24 '12 at 17:34